Some Results on the Composition of Distributions
DOI:
https://doi.org/10.5644/SJM.08.1.09Keywords:
Distribution, Dirac delta-function, Heaviside function, composition of distributions, neutrix, neutrix limitAbstract
Let $F$ be a distribution, $f$ be a locally summable function and $F_{n} = (F * \delta_{n})(x)$, where $\delta_{n}(x)$ is a certain sequence converging to the Dirac delta-function $\delta(x)$. The distribution $F(f)$ is defined as the neutrix limit of the sequence $\{ F_n(f) \}$, provided its limit $h$ exists in the sense that $$ \Nn \int_{-\infty}^{\infty} F_{n}(f(x))\varphi (x) dx = \la h(x) , \varphi (x) \ra $$ for all test functions $\varphi(x)$ in ${\mathcal D}(a,b)$. The composition of the distributions $x_{-}^{-s}\ln x_{-}$ and $x_{+}^{r}$ is evaluated for $r=0,1, \ldots$ and the composition of the distributions $x_{-}^{-s}\ln^{m} x_{-}$ and $H(x)$ is evaluated for $s,m=1,2,\ldots\,$.
2000 Mathematics Subject Classification. 46F10
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